Parallel Graph Drawing Algorithm for Bipartite Planar Graphs

TL;DR

This paper presents a parallel algorithm that efficiently assigns segments to vertices of bipartite planar graphs, enabling intersection-based representation while avoiding crossings, within logarithmic time on a CRCW PRAM.

Contribution

It introduces a novel parallel algorithm for segment assignment in bipartite planar graphs with optimal logarithmic time complexity.

Findings

Achieves $O( ext{log}(n))$ parallel time complexity.

Uses polynomial number of processors.

Ensures segments intersect iff vertices are adjacent.

Abstract

We give a parallel $O(\log(n))$-time algorithm on a CRCW PRAM to assign vertical and horizontal segments to the vertices of any planar bipartite graph $G$ in the following manner: i) Two segments cannot share an interior point ii) Two segments intersect if and only if the corresponding vertices are adjacent, which uses a polynomial number of processors. In other words, represent vertices of a planar bipartite graph as parallel segments, and edges as intersection points between these segments. Note that two segments are not allowed to cross. Our method is based on a parallel algorithm for st-numbering which uses an ear decomposition search.